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An osculating circle

In differential geometry of
curves, the osculating circle of a
sufficiently smooth plane curve
at a given point on the curve is the circle whose center lies on the inner normal
line and whose curvature is the same as that of the given
curve at that point. This circle, which is the one among all tangent circles at the given
point that approaches the curve most tightly, was named
circulum osculans (Latin for "kissing circle") by Leibniz.

The center and radius of the osculating circle at a given point
are called center of curvature and radius
of curvature of the curve at that point. A geometric
construction was described by Isaac Newton in his Principia:

There being given, in any places, the velocity with which a body
describes a given figure, by means of forces directed to some
common centre: to find that centre.

– Isaac Newton,
Principia; PROPOSITION V. PROBLEM I.

Contents

Description in lay terms

Imagine a car moving along a curved road on a vast flat plane.
Suddenly, at one point along the road, the steering wheel locks in
its present position. Thereafter, the car moves in a circle that
"kisses" the road at the point of locking. The curvature of the circle is equal to that of
the road at that point. That circle is the osculating circle of the
road curve at that point.

Mathematical description

Let γ(s) be a regular parametric curve, where s
is the arc length, or
natural parameter. This determines the unit tangent vector
T, the unit normal vector N, the signed curvaturek(s) and the radius of curvature at each
point:

Suppose that P is a point on C where
k ≠ 0. The corresponding center of curvature is the point
Q at distance R along N, in the same
direction if k is positive and in the opposite direction
if k is negative. The circle with center at Q and
with radius R is called the osculating
circle to the curve C at the point
P.

If C is a regular space curve then the osculating
circle is defined in a similar way, using the principal normal vectorN. It lies
in the osculating plane, the plane
spanned by the tangent and principal normal vectors T and
N at the point P.

Properties

For a curve C given by a sufficiently smooth parametric
equations (twice continuously differentiable), the osculating
circle may be obtained by a limiting procedure: it is the limit of
the circles passing through three distinct points on C as
these points approach P.[1]
This is entirely analogous to the construction of the tangent to a curve as a limit
of the secant lines through pairs of distinct points on C
approaching P.

The osculating circle S to a plane curve C at
a regular point P can be characterized by the following
properties:

The circle S passes through P.

The circle S and the curve C have the common tangent line at
P, and therefore the common normal line.

Close to P, the distance between the points of the
curve C and the circle S in the normal direction
decays as the cube or a higher power of the distance to P
in the tangential direction.

This is usually expressed as "the curve and its osculating
circle have the third or higher order contact" at P.
Loosely speaking, the vector functions representing C and
S agree together with their first and second derivatives
at P.

A circle with fourth order contact at a vertex of a curve

If the derivative of the curvature with respect to s is
nonzero at P then the osculating circle crosses the curve
C at P. Points P at which the derivative
of the curvature is zero are called vertices. If P is a vertex then
C and its osculating circle have contact of order at least
four. If, moreover, the curvature has a non-zero local maximum or minimum at P then
the osculating circle touches the curve C at P
but does not cross it.

The curve C may be obtained as the envelope of the one-parameter
family of its osculating circles. Their centers, i.e. the centers
of curvature, form another curve, called the evolute of C. Vertices of
C correspond to singular points on its evolute.